Geometry Proof (Circles and Chords)

[Volux]

Hey all. At my school we have these things called "Real Problems" that we get every 3-4 weeks. This time it's about proofs. I have an A- in the class, so I got the difficult one with 4 very difficult (IMO) proofs.

I have NO idea where to start on this..I have drawn it out for you. All help is greatly appreciated. I am here to merely learn, not copy everything. I would appreciate full answers, but I am definitely here to Learn how to do this for tests.

Given: Tangent line AB and chord BC.
Prove: <ABC = (1/2)(arc)BC

In other words prove: (Angle ABC = Half of arc BC)

HINT: Using the center of the circle, draw in OB and OC (which I did in the picture for you).

[PLAIN]http://img249.imageshack.us/img249/4413/proof1f.jpg

 

[mathwonk]

you might look in euclid where this is a standard theorem, probably in book III.

by the way you have probably started it more or less correctly, as the first step in a circle proof is always to draw an appropriate line to the center.

in fact the proof is immediate from your figure if you know the angle sum formula for a triangle, and the fact that a tangent is perpendicular to the radius.

 

[zgozvrm]

Consider drawing a line segment from the center of the circle, through the midpoint of BC, until it intersects AB.